Quadrants, the game

A 2 player game
played on an n-by-n grid, for n = odd.
(I SURE like making up games using grid gameboards, huh!?..)
Center grid-square has an x in it before play.
Players each have a fixed number of the directions they can play:
(and so each player should keep a tally of what they already have
played.)
Player 1 has:
(n^2 -1)/4 left's
(n^2 -1)/4 right's.
Player 2 has:
(n^2 -1)/4 up's
(n^2 -1)/4 down's.
Each player starts moving from the center, but thereafter moves from
the last position they personally placed an x in.
On each move, each player picks secretly one of their two (unexausted)
directions.
And then the 2 picked directions are revealed.
Each player next takes turns (player 1 then 2) putting an X into an
empty (un-X-ed) square which is located relative to the last square
they
played in the same (basic..*) relative direction as is indicated by
the 2 picked directions.
So, (by 'basic' relative direction, it is meant):
If____ is picked, players may 'move' to a square to the ___ of their
last played square.
left up ->
above & to left, or directly to left, or directly above
left down ->
below & to left, or directly to left, or directly below
right up ->
above & to right, or directly to right, or directly above
right down ->
below & to right, or directly to right, or directly below
So, play continues until a player has nowhere to move under the
conditions, and so the other player wins.
(If the grid is completely filled in with x's, this is a tie, but I
guess it is still an achievement in a way...)
thanks,
leroy Quet